We develop splitting techniques to study Lyubeznik numbers of cover ideals of
graphs which allow us to describe them for large families of graphs including
forests, cycles, wheels and cactus graphs. More generally we are able to
compute all the Bass numbers and the shape of the injective resolution of local
cohomology modules by considering the connected components of the corresponding
subgraphs. Indeed our method gives us a very simple criterion for the vanishing
of these local cohomology modules in terms of the connected components.

Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$
denote their fiber product over their common residue field $k$. We explore
consequences of vanishing of ${\rm Tor}^R_m(M,N)$ for small values of $m$,
where $M$ and $N$ are finitely generated $R$-modules.

In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich
sheaf on a general cubic hypersurface of dimension at most 7, whose existence
was proved by Manivel. Instead of using the invariant theory, we use Shamash's
construction with a cone over the spinor variety. We also describe an
algebro-geometric interpretation of our matrix factorization which connects the
spinor tenfold and the Cartan cubic.